368 APPENDIX: ELEMENTS OF LINEAR ALGEBRA

codim V or codimE V. If V has no supplementary subspace of finite dimen-
sion, then V is said to be of infinite codimension in E. By (A.4.5), V is of finite
codimension in E if and only if E is generated by the union of V and a finite
set of vectors.

(A.4.10) IfE is a finite-dimensional vector space, every subspace FofE is
finite dimensional and of finite codimension in E, and

(A.4.10.1) dim F + codim F = dim E.

For by applying (A.4.5) to F and a basis of E, it follows that there exists a
supplementary subspace F' of F in E, with dim F' < dim E. Interchanging
the roles of F and F', and using (A.3.5), it follows that F is also finite
dimensional If (xt\ ^ f^p is a basis of F and if (x^ ^j^q is a basis of F', it is
clear that the xt and the x'j together form a basis of E.

(A.4.11) Let E be a finite-dimensional vector space and let "F be a subspace of
E. If dim F = dim E, then F = E.

This follows immediately from (A.4.10.1).

(A.4.12) Let M and N be two finite-dimensional subspaces of a vector space E.
Then M + N is finite dimensional, and we have

(A.4.12.1) dim(M + N) + dim(M n N) = dim M + dim N.

The set consisting of the elements of a basis of M and a basis of N will
generate M + N, which is therefore of finite dimension (A.4.6). Let P (resp. Q)
be a supplementary subspace of M n N in M (resp. N). It is clear that
M -f N is the direct sum of M n N, P, and Q (A.3.3), and therefore
dim(M + N) = dim(M n N) + dim P + dim Q. But also (A.4.10) we have
dim P = dim M - dim(M n N), and dim Q = dim N - dim(M n N). Hence
the result.

(A.4.13) Let M and N be two vector subspaces of finite codimension in a vector
space E. Then M n N is of finite codimension in E, and we have

(A.4.13.1) codim(M + N) + codim(M n N) = codim M + codim N.umber of elements in a free system extracted from S is m < n. If (y^^ ^m